So I was trying to find some methods that would have impartial culture manipulability less than 100% of the time for four candidates as the number of voters approaches infinity. (I should strictly speaking say that I don't know if any methods stay below 100% here, but I had the impression that IRV does.) I did some brute forcing of methods of the type A's score = f(A) = sum over other candidates B != A: A>B * product over other candidates C, D not A nor B: H(x_1 * [fpA fpB fpC fpD]) * H(x_2 * [fpA fpB fpC fpD]) * H(x_3 * [fpA fpB fpC fpD]) highest score wins, with x_1, x_2, x_3 being vectors, and fpX being the first preference count of candidate X. This is a generalization of the Contingent vote, f(A) = sum over B A>B * product over C: H(fpA - fpB) * H(fpB - fpC) where H(x) as usual is the function H(x) = 0 if x < 0 1/2 if x = 0 1 otherwise As a simple monotonicity check I used the possibly too generous assumption that if we write the first term H(x_1 * [fpA fpB fpC fpD]) as H(x_11 fpA + x_12 fpB + x_13 fpC + x_14 fpD) then if x_11 >= max(x_12, x_13, x_14), then a voter who ranks say C first, can't harm A by raising A to top, because the benefit to the term inside the H can never decrease by doing so. So the method must be monotone since the A>B term is also monotone, hence the check x_11 >= max(x) is sufficient (but probably not necessary) for monotonicity. The results show a big nope: every such method (with the values of x integer between 2 and -2 inclusive) seem to be 100% manipulable in the limit. However, it did come up with this, which seemed to converge less quickly, and might be an interesting idea on its own: f(a) = sum (or max) over B: A>B * H(fpA + fpB - fpC - fpD) I.e. for four candidates, there'll always exist at least one pair of candidates who have more than majority combined first preference support (unless there's a four-way tie). Give each candidate in this pair his pairwise defeat strength against the other. (In the case where there are multiple, e.g. a single candidate has majority, using sum would give a different method than using max.) -km

"To boldly go where no man has gone before!" It takes a very tenacious mind... with bulldog tenacity... to pursue an idea this far into the wild unknown! On Wed, Dec 21, 2022, 10:18 AM Kristofer Munsterhjelm <km_elmet@t-online.de> wrote: > So I was trying to find some methods that would have impartial culture > manipulability less than 100% of the time for four candidates as the > number of voters approaches infinity. (I should strictly speaking say > that I don't know if any methods stay below 100% here, but I had the > impression that IRV does.) > > I did some brute forcing of methods of the type > > A's score = f(A) = sum over other candidates B != A: > A>B * > product over other candidates C, D not A nor B: > H(x_1 * [fpA fpB fpC fpD]) * > H(x_2 * [fpA fpB fpC fpD]) * > H(x_3 * [fpA fpB fpC fpD]) > > highest score wins, with x_1, x_2, x_3 being vectors, and fpX being the > first preference count of candidate X. This is a generalization of the > Contingent vote, > > f(A) = sum over B > A>B * > product over C: > H(fpA - fpB) * H(fpB - fpC) > > where H(x) as usual is the function > H(x) = 0 if x < 0 > 1/2 if x = 0 > 1 otherwise > > As a simple monotonicity check I used the possibly too generous > assumption that if we write the first term > H(x_1 * [fpA fpB fpC fpD]) > as > H(x_11 fpA + x_12 fpB + x_13 fpC + x_14 fpD) > then if x_11 >= max(x_12, x_13, x_14), then a voter who ranks say C > first, can't harm A by raising A to top, because the benefit to the term > inside the H can never decrease by doing so. So the method must be > monotone since the A>B term is also monotone, hence the check x_11 >= > max(x) is sufficient (but probably not necessary) for monotonicity. > > The results show a big nope: every such method (with the values of x > integer between 2 and -2 inclusive) seem to be 100% manipulable in the > limit. However, it did come up with this, which seemed to converge less > quickly, and might be an interesting idea on its own: > > f(a) = sum (or max) over B: > A>B * H(fpA + fpB - fpC - fpD) > > I.e. for four candidates, there'll always exist at least one pair of > candidates who have more than majority combined first preference support > (unless there's a four-way tie). Give each candidate in this pair his > pairwise defeat strength against the other. > > (In the case where there are multiple, e.g. a single candidate has > majority, using sum would give a different method than using max.) > > -km > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info >

On 12/21/22 21:54, Forest Simmons wrote: > "To boldly go where no man has gone before!" > > It takes a very tenacious mind... with bulldog tenacity... to pursue an > idea this far into the wild unknown! As the bulldog, I'm reminded of Barry Mazur's quote about number theory: "number theory swarms with bugs waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort!". Something like that, but not number theory in my case :-) I've investigated a bit further, and it seems like every method (that's manipulable to begin with) becomes manipulable with probability 100% with enough voters and the impartial culture model. If I were to guess, I'd say this has to do with the the margins A>B-B>A approach zero (assume A beats B pairwise wlog). So it's very easy to overturn a pairwise victory. Perhaps that could be used to show that you can always make someone else the CW, and then use Durand's result to show that other majority methods can be no better than Condorcet methods at resisting strategy. Anyway... with a limited number of voters, quadelect shows that the best monotone method is not far off the best nonmonotone (only 6% more manipulable for 97 voters). But I can't reproduce this with my simple Python simulator. So one of the two, or both, are wrong, and that needs further examination... ... which I probably won't get around to this year. There ought to be time for relaxing and enjoying holidays too! Not just satisfying one's curiosity. (Also, datapoints from others' simulators would be very useful. In particular, what's the manipulability of the "two candidates share a majority, then A>B" method of my last post?) -km